I am making a small paperweight liquid toy, and I have no idea which type of paint to use to paint the buoyant object inside in order for it to not lose its colours while moving inside.
This is an example:
If I use water based paint on the boat object (the same paint that is used to paint on paper), would it fade away or wash out because it is moving inside the oil and water?
Water and oil are both problem liquids. Water-based finishes (not water colors, which are inherently unsuitable, but acrylics, for example), improve their water resistance as they dry and cure, but they can often degrade if submerged in water for a very long time. Oil can affect the adhesion of the paint, and can break down some finishes over very long exposure. The model repeatedly moving through the liquid would also help to strip off any degraded finish.
The image used in the example looks like the colors are in layers. If that will mostly be the case, you could cast the model in layers using colored resin, which would avoid the problems of adding a finish.
If there are areas that don't lend themselves to that, you could paint those with colored resin of the same type, or another resin or adhesive designed to bond with it (you can add colorants to resins and adhesives). Use a material that integrates with the model, or forms a bond as strong as the resin, itself.
Using the same resin is ideal; otherwise, think in terms of materials that will weld themselves to the surface and can be made the color you want, rather than looking at paint or finishes designed to be added decoration. For example, there are some epoxies that are formulated to form a really strong bond with many plastic resins.
Use proper surface preparation. It can also help to include shallow recesses in the mold for the "paint" spots so that the paint spot doesn't have a raised edge.
Acrylics should do the trick and you can seal it with a polyurethane varnish to help protect the colors from wearing away - it also gives a lovely glossy finish.
Look up tutorials on painting and sealing polymer clay, they should give you more in depth info if you need.
Water colours will certainly wash off, if you are able to apply them in the first place.
As suggested in the comments, acrylics might work, but that likely depends on the bond with the substrate. You can ground the plastic, and/or roughen it up slightly with some sandpaper. Acrylics will need a coating or sealer to make it waterproof.
